There are a lot of cheap flights available within the European Continent. The real issue is how much time the train takes vs. flying. For example, the train from Berlin to Paris takes roughly 8 hours non stop.  Add an hour to get to the train station and then 30 minutes to get to your hotel. Compare the 9 1/2 hours by train to an hour to get to the airport plus arriving 90 minutes in advance (assuming no checked luggage), a 2 hour flight, and one hour for travel to your hotel. That is 5 1/2 hours. Unless there is a significant cost difference, flying makes the most sense. However London to Paris is about the same:  4 1/2 by plane vs. 4 hours by train. One other consideration is how relaxing is it on the train vs. flying?

• TIP:  Non-American airlines have different size restrictions for carry-on luggage. While you may not be affected by this flying from the US to Europe, you might be forced to check your bags when flying intra-Europe or on the return flight back to the US.